An application of relative difference quotient algorithm to topology optimization of truss structures with discrete variables

Discrete sizing and topology optimization of truss structures subject to stress and displacement constraints has been formulated as a Mixed-Integer Linear Programming (MILP) problem. The computation time to solve a MILP problem to global optimality via a branch-and-cut solver highly depends on the problem size, the choice of design variables, and the quality of optimization constraint formulations. This paper presents a new formulation for discrete sizing and topology optimization of truss structures, which is benchmarked against two well-known existing formulations. Benchmarking is carried out through case studies to evaluate the influence of the number of structural members, candidate cross sections, load cases, and design constraints (e.g., stress and displacement limits) on computational performance. Results show that one of the existing formulations performs significantly worse than all other formulations. In most cases, the new formulation proposed in this work performs best to obtain near-optimal solutions and verify global optimality in the shortest computation time.

A new fuzzy strategy for size and topology optimization of truss structures

This paper addresses single and multiobjective topology optimization of truss-like structures using genetic algorithms (GA's). In order to improve the performance of the GA's (despite the presence of binary topology variables) a novel approach based on kinematic stability repair (KSR) is proposed. The methodology consists of two parts, namely the creation of a number of kinematically stable individuals in the initial population (IP) and a chromosome repair procedure. The proposed method is developed for both 2D and 3D structures and is shown to produce (in the single-objective case) results which are better than, or equal to, those found in the literature, while significantly increasing the rate of convergence of the algorithm. In the multiobjective case, the proposed modifications produce superior results compared to the unmodified GA. Finally the algorithm is successfully applied to a cantilevered 3D structure.

Topology optimization of truss structures is considered in this paper. Trusses are widely used in various constructions: bridges, towers, roof supporting structures. Topology optimization of trusses requires large amount of computing resources. Therefore distributed computer networks are used to solve this kind of problems. In this paper a distributed branch-and-bound combinatorial algorithm for topology optimization of trusses is investigated in computational grid infrastructure and clusters of personal computers.

In this paper, a technology enabling to optimize the topology of truss or frame structures with genetic algo-rithms is presented. The objective function is total mass of the structure elements. Constrains consist of equilibrium equations and conditions on stresses in the elements: stresses cannot exceed allowable value and must be above some threshold value. The values of the objective function are calculated with package ANSYS. Genetic algorithm with interface is implemented in C++. Exchange of data between ANSYS and C++ goes through temporary files. Numerical examples are presented.

Correction: MILP-based discrete sizing and topology optimization of truss structures: new formulation and benchmarking

Explicit control of 2D and 3D structural complexity by discrete variable topology optimization method

During the last 20 years, structural optimization has become one of the most important topics of engineering applications. Design optimization of structure has been an interesting area of research in the field of engineering design for its ability to short the design cycle and to enhance product quality. Significant research activity has occurred in the area of structural optimization in the last decade. Especially for topology optimization of structure, many new theoretical, algorithmic, and computational contributions have resulted by researchers and engineers. Topology optimization is a powerful tool for global and multi-scale design of macrostructures, microstructures, and the cell of prescribed composite materials. The population based evolutionary algorithms have emerged as powerful mechanism for finding optimum solutions of complex optimization problems in engineering during the last two decades. Evolutionary computation is the study of computational systems which use ideas and get inspiration from natural evolution and adaptation [1]. The thinking has wide application in various engineering fields, such as computer science, artificial intelligence, operations research. Genetic algorithm is another kind of bio-inspired optimization method and it is playing an increasingly important role in studies of complex adaptive systems. Its application ranges from adaptive agents in economic theory to the use of machine learning techniques in the design of complex devices and structures, such as aircraft turbines and integrated circuits [2]. Optimization of structures can be classified into three categories: sizing, shaping, and topology optimization. In the topology optimization, it is concerned with the structure members and connectivity between members. In general, it is easily represented by discrete variables rather than by those used for continuous optimization problems. Topology optimization is the most difficult and complex among three categories and it is special useful in developing innovative conceptual designs. Structural optimization, in particular the topology optimization, has been identified as one of the most challenging tasks in structural design. Various techniques and approaches have been established during the last two decades. Topology optimization usually referred to as layout optimization or general shape optimization [3]. It lets engineers get the optimal topology of structure or new configurations during product design phase, as they are implementing the design of the size

Stress-based topology optimization and nonlinear structural topology optimization is gaining increasing attention in order to make topology optimization more realistic. Thus, this paper extends current concepts of topology optimization to the design of structures made of nonlinear materials. An extended bi-directional evolutionary structural optimization (BESO) method for stress minimization topology optimization of material nonlinear structures is proposed in this work. BESO method based on discrete variables can effectively avoid the well-known singularity problem in density-based methods with low-density elements. The maximum von Mises stress is approximated by the p-norm global stress. The sensitivity information for designing variable updates is derived in detail by adjoint method. As for the highly nonlinear stress behavior, the updated scheme takes advantages from two filters respectively of the sensitivity and topological variables to improve convergence. Moreover, the filtered sensitivity numbers are combined with their historical sensitivity information to further stabilize the optimization process. The effectiveness of the proposed method is demonstrated by several 2D benchmark design problems.

This paper will give a short survey about topology optimization of structures. It is particularly focused on topological shape optimization of structures using level-set methods, including the level-set based standard methods and the level-set based alternative methods. The former often directly solve the Hamilton-Jacobi partial differential equation (H-J PDE) to obtain the boundary velocity field using Finite Differential Methods (FDM), and the later commonly employ parametric or equivalent methods to evaluate the velocity field without directly solving the H-J PDE. The unique characteristics of the level-set based topology optimization methods are discussed, and a future perspective and prospects in this research area is also included. A benchmark numerical example is used to showcase the effectiveness of the level-set based methods.

Dimension reduction and surrogate based topology optimization of periodic structures

A novel optimization method based on a differential evolution (DE) strategy for designing low-weight truss structures is presented for both continuous and discrete variables. Applications of this technique on the optimization of a benchmark-type truss structure with continuous and discrete variables are given to evaluate its effectiveness. Results are compared with various classical and evolutionary optimization methods which show that the proposed procedure based on DE outperforms other methods and can effectively be applied to the optimization problems of truss structures with both continuous and discrete design variables.

Structural topology optimization seeks to identify the most efficient way to allocate material within a given domain to achieve a specific design objective. This research area has broad applications in various fields, including mechanical design and other physical disciplines. Over the past two decades, a great variety of applications and diverse research papers on structural topology optimization have been undertaken and published in the last two decades to highlight its acceptance and advantages. This paper presents a scientometric analysis of 5713 papers published between 2000 and 2022 to provide an overview and visualization of the development of structural topology optimization. While there are numerous approaches to topology optimization, this study focuses on presenting a quantitative analysis of the most popular and influential methods. The method used in this paper involved data collection and analysis through CiteSpace, one of the most popular software tools in this field. This paper provides a comprehensive overview of the latest developments of research in structural topology optimization and identifies key directions for future research. The analysis of the top 10 co-cited articles revealed four main approaches that have been developed for structural topology optimization.

This article proposes an efficient parameterized level set method (PLSM) for structural topology optimization design with minimum compliance as the objective function and volume fraction as the constraint condition. In the stage of establishing the optimization model, the level set function (LSF) is interpolated using compactly-supported radial basis functions (CS-RBFs) to transform the Hamilton-Jacobi partial differential equation (PDE) into ordinary differential equations (ODEs), thereby making the evolution of the LSF more convenient and efficient, and ensuring the smoothness of the optimization result boundary. The method of moving asymptotes (MMA) is used for solving the established optimization model, and meanwhile the shape sensitivity constraint factor is added to improve computational efficiency. During the evolution of the LSF, an approximate re-initialization scheme is employed to prevent the gradient of the LSF boundary from being too large or too small, thereby improving the numerical stability and the convergence speed of structural topology optimization process. Furthermore, the proposed method is also extensible and applicable to topology optimization of multi-material structures. The feasibility and effectiveness of this method have been verified through several typical numerical examples involving topology optimization of single-material structures and multi-material structures within the framework of minimum compliance design. HIGHLIGHTS An efficient parameterized level set method using the approximate re-initialization scheme is proposed for structural topology optimization. The approximate re-initialization scheme is employed when solving the parameterized level set function via the MMA algorithm. This scheme can prevents the gradient of the level set function boundary from being too large or too small, making the level set function update more stable and accelerating the convergence speed of structural topology optimization. The effectiveness and feasibility of the proposed method are demonstrated through examples of single-material and multi-material structural topology optimization.

This thesis aims at understanding and improving topology optimization techniques focusing on density-based level-set methods and geometrical nonlinearities. Central in this work are the numerical modeling of the mechanical response of a design and the consistency of the optimization process itself. Concerning the first topic, we investigate different means to improve the robustness of density-based numerical models including geometrical nonlinearities. The conventional approach (scaling the local material properties) can result in convergence problems due to excessive deformation in low-stiffness finite elements. To avoid excessive deformation, we combine the element connectivity parameterization method (adapting the connectivity between finite elements) with level-set-based topology optimization. Furthermore, we achieve greater robustness of analysis method using a second and improved approach called element deformation scaling. This approach eliminates the need for solving internal equilibrium equations (needed for the element connectivity parameterization method) via an explicit relation between the local internal and global external displacement field. The second focus of this thesis is on the optimization process of level-set-based topology optimization, and in particular its numerical consistency. We observe that signed-distance reinitialization of the level-set function affects the shape of a design in the optimization process. To minimize this effect, we propose a discrete level-set method that is based on an approximate Heaviside function and focusing on the implementation. Furthermore, we also propose a level-set-based topology optimization method using an exact Heaviside function and mathematical programming that effectively eliminates the need for reinitialization. We demonstrate that our density-based level-set method is closely related to conventional density-based topology optimization methods, while offering the advantage of more control over the geometrical complexity. On the other hand, we confirm that the dependence of the final result on the initial design which remains one of the big challenges for level-set-based topology optimization. The potential of the proposed level-set method is shown by applying it on problems with stress constraints and geometrical nonlinearities and performing manufacturing tolerant topology optimization. Finally, this thesis offers a review of level-set methods for structural topology optimization to identify and discuss the different approaches that are available in literature. We can distinguish between level-set methods by examination of their design parameterization, sensitivities, update procedures and regularization techniques. A level-set-based design parameterization offers the advantage of a crisp distinction between subdomains. For this reason, X-FEM approaches and conforming discretizations are an interesting option to retain the crisp nature of the level-set-based description of the design. Many level-set methods are combined with density-based numerical models and are, therefore, closely related to conventional density-based topology optimization methods. In particular, recently proposed projection methods have much in common with a level-set-based design description. The results of level-set-based TO methods often rely heavily on regularization techniques that introduce inconsistencies in the optimization process. Numerical consistency does not necessarily lead to the best search direction, but is essential to find a Karush-Kuhn-Tucker point of the discretized optimization problem.